In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claim …

This should really be well-known, but I was not able to find a definite answer to this question: Is the Fourier transform of a bounded function always a borel measure (i.e. an ord …

The Fourier Transform is in general terms a continuous mapping, as well as it's inverse when it exists. Is there any abstract result that translates convergence of the transforms …

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range is …

Hi! After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least …

I would like to know if there's a way to compute or approximate the following expectation: $$\mathbb{E}[(c+e^X)^{-n}]$$ where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume t …

Hey guys, I'm currently doing a course in stochastic processes and have come across something that has been wrecking my mind for a while. So, let's say that I have some even, symm …

What is the solution of the fractional differential equation $$ f^{(\alpha-1)}(t) = tf(t) $$ where $(\alpha)$ denotes the fractional derivative of order $\alpha$ EDIT: Background …

How do you calculate the inverse Fourier transform of $\frac{k_ik_j}{k^4}$. I know it has to be a matrix of the form $=δ_{ij}A(r)+r_ir_jB(r)$, but how do you calculate the function …

For a positive function $f(x)$, its auto correlation function $A(x)=\int_{-\infty}^{\infty} f(s)f(s+x) ds>0$ and is positive-definite (its Fourier transform $\mathcal{F}(A(x))>0$). …

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heurist …

Dear all, please help me solve the following integral. I need to solve this integral for one of my problems. $$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho …

Let $\widehat{f}(\xi)$ be Fourier transform of $f$ given by \begin{align} \widehat{f}(\xi)=\int_{\mathbb{R}^n} e^{-ix\cdot\xi}f(x)dx. \end{align} Suppose that $\widehat{f}(\xi)$ …

The free resolvent in $\mathbb{R}^3$ has this rapresentation: $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$ with $\Im\sqrt{z}>0$. So its integral k …

I am studying Fourier transforms by watching Stanford online videos on youtube. I am trying to solve the following problem but i do not know how to relate it to poisson summation f …